Gravitational instantons with source

We discuss various geometric formulations of the equations of gravitational instantons with sources. We show that these equations include as a special case the classical Einstein equations with or without the cosmological constant. We discuss the notion of an Einstein pair to clarify the distinction between the Palatini and the metric variational equations. Our approach naturally leads to the introduction of a cosmological function and a generalized conservation law for the energy-momentum tensor. We also discuss the physical significance of the new conservation condition